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NUMBER SYSTEMS Number Theory Is the Study of the Integers. We NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = f:::; −3; −2; −1; 0; 1; 2; 3;::: g: The integers have two operations defined on them, addition and multi- plication, which are associative (a + (b + c) = (a + b) + c, a(bc) = (ab)c for a; b; c 2 Z) and commutative (a + b = b + a, ab = ba). Moreover, these operations interact via the distributive law (a(b + c) = ab + ac) and have neutral elements 0 and 1 respectively (a+0 = a and a·1 = a). Notice also that each integer can be negated (a+(−a) = 0). In modern algebra language, a set having the aforementioned properties is called a commutative ring. The two operations in Z are not, however, \created equal", for while every integer can be negated (for example, 3 + (−3) = 0), not every integer can be inverted (there is no integer b such that 3b = 1). Indeed, the only integers whose reciprocals are also integers are 1 and −1. In general, an element a of a commutative ring is called a unit if there is an element b of the ring such that ab = 1. The rational numbers, denoted by Q, are all the ratios of integers: na o = : a; b 2 and b 6= 0 Q b Z 4 2 (of course, we consider 6 and 3 , for example, to be the same element of Q). Like Z, Q is a commutative ring, but in contrast any nonzero ele- a b ment of Q is a unit (if b 2 Q and a 6= 0, then a 2 Q also); commutative rings having this additional property are called fields. Another field you are familiar with is the real numbers, denoted by R. Notice that Z ⊂ Q ⊂ R. We know how Z sits in R (imagine a number line with the integers marked off); you may have thought less about how Q sits in R. Proposition 1. Between any two real numbers, there is a rational number. Proof. Suppose a; b 2 R with a < b. Let n be a positive integer large 1 enough that n < b − a. Since the rational numbers f :::; −2=n; −1=n; 0; 1=n; 2=n; : : : g 1 are spaced n apart, at least one of them lies between a and b. Because of Proposition 1, we say that Q is dense in R. However, not every real number is rational; a real number which is not rational is called irrational. Proposition 2. e is irrational. Proof. Suppose that e were rational. Then e = a=b for some positive integers a and b. It follows that the number α defined by 1 1 1 1 α = b! e − 1 − − − − · · · − 1! 2! 3! b! is an integer (imagine multiplying the b! through). Moreover, since e is defined by 1 X 1 e = ; n! n=0 we also have that α is positive. Next note that the definition of e, along with the formula for the sum of a convergent geometric series, implies that 1 1 1 1 α = b! + + ··· = + + ··· (b + 1)! (b + 2)! b + 1 (b + 1)(b + 2) 1 1 1 b+1 1 < + 2 + ··· = 1 = ≤ 1: b + 1 (b + 1) 1 − b+1 b We conclude that α < 1, which is a contradiction since α is a positive integer. Note that e is an infinite sum of positive rational numbers - as such, it is the limit of an increasing sequence of rational numbers (namely, the sequence of partial sums) - yet e itself is not rational. The set of real numbers R has the remarkable property that every increasing sequence of rational numbers is either unbounded or converges to an element of R. In fact, R is the smallest such field, in the sense that any other field which contains Q and has this property also contains R as a subfield. We will see more irrational numbers later; in fact, it turns out that the irrationals are much more numerous than the rationals. Another field that you may have worked with is the field of complex numbers C: C = fa + bi j a; b 2 Rg; where i2 = −1. Many of the commutative rings that we study in these notes (for example, Z, Q and R) are contained in C. DIVISIBILITY Let us first focus on the multiplicative structure of Z. We begin by discussing how integers break down into simpler multiplicative parts. Definition 3. If a; b 2 Z, we say that b divides a, and write b j a, if there is an integer c such that a = bc. Synonyms for \b divides a" that you may be familiar with are \b is a divisor of a", \b is a factor of a", \a is a multiple of b" and \a is divisible by b". If b is not a divisor of a, we write b - a. Example 4. 3 j 12, 7 - 16 Example 5. The positive divisors of 30 are 1; 2; 3; 5; 6; 10; 15 and 30. Notice that any integer a is a divisor of 0 (0 = a · 0) and is divisible by 1 (a = 1 · a). A direct consequence of the former statement is the following surprisingly useful result. Corollary 6. If a is an integer and there is a positive integer b such that b - a, then a 6= 0. We have thus far only discussed divisibility in Z. The analogous notion of divisibility in Q is trivial in the following sense: if r is a nonzero rational number, then r divides every rational number (this follows from the fact that we can invert any nonzero element of Q). In fact, the same is true in every field; for this reason, when we discuss di- visibility we will mean it in the context of the integers unless otherwise stated. Proposition 7. Let a; b; c 2 Z. (1) If a j b and b j c, then a j c. (2) If a j b and a j c, then for any integers x and y, a j (xb + yc). Proof. (1) Since a j b and b j c, there are integers m and n such that b = am and c = bn. Then c = (am)n = a(mn). Since mn is an integer, it follows that a j c. (2) Since a j b and a j c, there are integers m and n such that b = am and c = an. Then xb + yc = x(am) + y(an) = a(xm + yn); and so a j (xb + yc). THE PRIMES Notice that every integer n > 1 has at least two positive divisors, namely 1 and n (these are sometimes called the trivial divisors of n). If d j n and 1 < d < n, d is called a proper divisor of n. Definition 8. An integer p > 1 is called prime if its only positive divisors are 1 and p (i.e., if it has no proper divisors). An integer n > 1 that is not prime is called composite. Example 9. The first five primes are 2; 3; 5; 7 and 11. Primes can therefore be thought of as multiplicatively the simplest positive integers. We now establish their central place in multiplicative number theory. Proposition 10. If an integer n > 1 is composite, then the smallest proper divisor of n is prime. Proof. Let d be the smallest proper divisor of n. If d had a proper divisor m, then m would be a divisor of n by Proposition 7 (1), and since 1 < m < d < n, m would be a proper divisor of n. Since m < d, this contradicts that d is the smallest proper divisor of n. Therefore d has no proper divisors, i.e., d is prime. Theorem 11. Every integer n > 1 is a product of primes. Proof. By induction. Since 2 is prime, it is the product of a single prime, so the statement holds for n = 2. Now suppose it holds for all the integers from 2 up to n. If n + 1 is prime, the statement holds for n + 1. If n + 1 is composite, then by Proposition 10 it has a proper prime divisor p. Write n + 1 = pm. Since 1 < p < n + 1, it follows that 1 < m < n + 1, i.e. 2 ≤ m ≤ n. By the induction hypothesis m is a product of primes, and therefore so is pm = n + 1. Example 12. 84 = 2 · 42 = 2 · 2 · 21 = 2 · 2 · 3 · 7 We see that the primes are the multiplicative building blocks of Z, and therefore it is natural to study them as a distinguished set. One natural question to ask is \how many primes are there?" Theorem 13. (Euclid) There are infinitely many primes. Proof. Let S be any nonempty finite set of primes. Consider the integer Y n = 1 + p: p2S If n is prime, then since n is larger than any element of S, we have that n2 = S. If n is composite, then by Proposition 10 it has a prime divisor q. Notice that q2 = S, for if q were an element of S, then it Q would divide p2S p = n − 1, and then by Proposition 7 (2) it would divide 1 · n + (−1) · (n − 1) = 1, a contradiction. We see that in all cases, there is a prime that lies outside S. It follows that no finite set of primes contains every prime, and thus the set of primes is infinite. Let us now consider the problem of identifying the primes among the positive integers. Suppose we start from the very definition of a prime: an integer p > 1 with no proper divisor.
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